Line Equation

General line equation

To set up the straight line, you only need the slope and the intersection of the straight line with the $y$-axis.

In this equation, $\textcolor{ff6600}{m}$ is the slope of the straight line and $\textcolor{ff6600}{m}$ is the $y$-value where the straight line intersects the $y$-axis.

Constituents of a line equation

A straight line equation consists of a gradient and the $y$-axis intercept $t$. These components are explained in more detail below.

Slope/Gradient

The slope indicates how steep a straight line climbs or falls. From the given equation, one can read out the gradient $m=2$.

If you did not know this, you can also determine the gradient using a gradient triangle. To do this, you need at least two different points, which you can obtain by inserting different $x$-values.

The $y$-axis intercept $t$

The $y$-axis intercept $t$ indicates the $y$-value at which the straight line intersects the $y$-axis. The value is also obtained by inserting zero for $x$ into the equation of the straight line, since $m\cdot x$ is omitted for the case $x=0$ and only $y=t$ remains from the original equation.

The fact that the $y$-axis intercept $t$ has the value 3 in the example can also be seen in the drawing from the line intersecting the $y$-axis at point $B$. $B$ has the coordinates $\left(0\left|3\right.\right)$.

Calculate the equation of a line through two points

Example: Given are the points $A(-1|1)$ and $B(2|3)$. Calculate the equation of the straight line that passes through $A$ and $B$.

1. Calculate the slope using the difference quotient              $m=\frac{\;\;1-3}{-1-2}=\frac{-2}{-3}=\frac23$

2. Plug $m$ and any point into the equation of the line to determine $t$. We use the point $B$.

   $\def\arraystretch{1.25} \begin{array}{l}\begin{array}{ccccc}\mathrm y&=&\mathrm m\cdot\mathrm x+\mathrm t&&\\3&=&\frac23\cdot2+\mathrm t&\;\;\;\;\;\;\;\;\;\;\left|-\frac43\right.&\;\\3-\frac43&=&\mathrm t\;\;\;\;\;\;\;\;&\;\;&\;\end{array}\\\begin{array}{ccc}\;\;\;\;\;\;\;\mathrm t&=&\frac53\;\;\;\;\;\;\;\end{array}\\\end{array}$

3. Substitute $m$ and $t$ into the general equation of a line. $\;\;\Rightarrow\;\;\mathrm y=\frac23\mathrm x+\frac53$

Calculate the equation of the straight line given the $y$-axis intercept $t$ and a point $P$.

Example: Given are the $y$-axis intercept $t =-3$ and the point $P(2/1)$. Calculate the corresponding equation of a straight line.

1. Insert $t$ and the coordinates of the point $P$ into the general line equation and solve for $m$.

2. Substitute $m$ and $t$ into the general equation of a straight line $\Rightarrow \;\;y=2x-3$

General lines (interaktive)

Special lines